Like the ZX- and ZW-calculi, the ZH-calculus includes a family of “Z-spiders” associated with the computational basis. However, its point of departure is the inclusion of “H-boxes”, which are \(n\) -ary generalisations of the Hadamard gate satisfying a variation of the spider fusion law, much like the one satisfied by \(W\) -spiders in the ZW-calculus.Despite satisfying a similar variation of the spider fusion rule, this generalisation of the Hadamard node is different from that employed in the original version of the Y-calculus [1]}. Whereas Hadamard gates are used to introduce edges between 2 vertices in a graph state, H-boxes can introduce hyperedges between \(n\)  vertices in a hypergraph state. Seen from another perspective, H-boxes are closely related to both \(n\) -ary AND gates in classical logic and to the Hadamard-CCZ family of quantum circuits.
As a result, Boolean circuits can be encoded in the ZH-calculus with low constant overhead. In particular, the linear maps corresponding to classical AND and NOT gates can be depicted as follows in terms of the ZH calculus:
logic
While the unitary NOT gate has a simple expression in the ZX-calculus, a typical encoding of an AND gate requires 25 basic generators and non-Clifford phases (cf. [2]}, §12.1.3.1).
Similarly, multiply-controlled phase gates also have very succinct representations, indicating that the ZH-calculus may be useful for analysing Hadamard-CCZ circuits (a.k.a. Hadamard-Toffoli circuits [3]}, [4]}, cf. forthcoming [5]} for connection to ZH), as well as diagonal operators at any level of the Clifford hierarchy [6]}.
